Tuesday, 27 Sep 2022
Education Online

Horizontal Asymptotes: Definition, Calculation, Rules

Before we go into the concept of a horizontal asymptote, let’s define a function. A function is an equation that shows how two things are related. Typically, functions explain how y is related to x. Functions are frequently graphed to offer a visual representation.

What are horizontal asymptotes?

A horizontal asymptote is a horizontal line that shows how a process will behave at the graph’s extreme edges. A horizontal asymptote is not hallowed ground. The function can get close to, and even cross, the asymptote. Horizontal asymptotes occur for functions with polynomial numerators and denominators. These are known as rational expressions.

Let’s take a look at one to see how a horizontal asymptote appears. As a result, our function is a product of two polynomials. y = 0 is our horizontal asymptote. Examine how the function’s graph approaches the ends of the graph and grows closer and closer to that line. We can plot some points to examine how the function behaves at extremes. Do you see how, at the extremely far margins, the function comes closer and closer to the line y = 0? If a function has a horizontal asymptote, this is how it behaves around it. Horizontal asymptotes do not exist for all rational formulations. Firstly, let us discuss the rules of horizontal asymptotes to see when and how a horizontal asymptote will occur.

If you want to learn more on this topic, keep reading. We will be shedding ample light on horizontal asymptotes.

                                                                         

Horizontal Asymptotes Definition

The horizontal asymptotes of a function reflect the values of f(x) when x is considerably small or notably big. They also indicate the function’s value as x- infinite. The horizontal asymptotes of a function reflect the values of f(x) when x is quite small or notably big. They also indicate the function’s value as x- infinite. An asymptote of a curve is a line that is tangent to the curve at infinity in projective geometry and similar settings. Asymptotes are classified into three types: horizontal, vertical, and oblique. Vertical asymptotes are vertical lines where the function increases indefinitely. An oblique asymptote has a non-zero but finite slope. Asymptotes provide information about the large-scale behaviour of curves. As a result, finding a function’s asymptotes is a crucial step in drawing its graph.

  • Horizontal asymptotes frequently denote a point beyond which the y values cannot be increased any further.
  • There is a well-known counter-argument to the Malthusian population curve. It employs horizontal asymptotes to depict a point beyond which the population cannot continue to expand.
  • A horizontal asymptote also generally represents a supremum of infimum.
  • One thing to consider is how your asymptotes influence the change in your dependent variable (whatever that might be). They must vary in response to the changes in your independent variable.

Caution:

 A widespread misconception among students is that a graph cannot cross a slant or horizontal asymptote. This is not true! A graph can have both vertical and horizontal asymptotes (sometimes more than once). A graph cannot pass those vertical asymptote creatures. This is because they are the domain’s weak points.

 Horizontal Asymptotes Calculation

To begin, keep in mind that the denominator is a sum of squares. Therefore it does not issue and contains no actual zeroes. In other words, there are no vertical asymptotes for this logical characteristic. So we’re good on that front. As previously said, the horizontal asymptote of a feature tells us roughly where the graph will be heading. This is because x is actually genuinely huge. So we’ll look at a couple of really big x numbers. At a few x values, that can be extremely far from the origin.

An asymptote is also a line that a function’s graph approaches but never touches. As shown in this question, rational functions have asymptotes. There is a vertical asymptote at x = 3 and a horizontal asymptote at y = 1 in this case. The curves approach but never cross these asymptotes. Depending on the degree, the method used to locate the horizontal asymptote varies. The polynomials shape the function in the numerator and denominator.

  • Divide the coefficients of the highest degree terms if both polynomials are of the same degree.
  • If the polynomial in the numerator has a lower degree than the denominator, the x-axis (y = 0) is the horizontal asymptote.
  • There is no horizontal asymptote if the polynomial in the numerator has a greater degree than the polynomial in the denominator. There is a slant asymptote, which we will look at in a later session.

                                                                          Horizontal Asymptotes

Horizontal Asymptotes Function

Before we go into the concept of a horizontal asymptote, let’s define a function. A function is an equation that shows how two things are related. Typically, functions explain how y is connected to x. Functions are frequently graphed to offer a visual representation. A horizontal asymptote is a horizontal line that indicates how a procedure will behave at the graph’s extreme edges. A horizontal asymptote, on the other hand, is not hallowed ground. The function can get close to, and even cross, the asymptote. Horizontal asymptotes occur for functions with polynomial numerators and denominators. These are known as rational expressions. Let’s take a look at one to see what a horizontal asymptote looks like.

As a result, our function is a product of two polynomials. y = 0 is our horizontal asymptote. Examine how the function’s graph approaches the ends of the graph and grows closer and closer to that line. We can plot some points to examine how the function behaves at its extremes.

X Y
-10,000 -0.0004
-1000 -0.004
-100 -0.04
-10 -0.4
-1 -4
1 4
10 0.4
100 0.04
1000 0.004
10,000 0.0004

Do you see how, at the extremely far margins, the function comes closer and closer to the line y = 0? If a function has a horizontal asymptote, this is how it behaves around it. Horizontal asymptotes do not exist for all rational formulations. Let us now discuss the rules of horizontal asymptotes to see when and how a horizontal asymptote will occur.

Horizontal Asymptotes Rules

Horizontal asymptotes follow three rules, depending on the degree of the polynomials included inside the rational expression.

Our feature has a polynomial of degree n at the top and a polynomial of degree m at the bottom. Our horizontal asymptote rules are entirely based on these stages.

  • When n is much less than m, the horizontal asymptote is y = zero, often known as the x-axis.
  • Also, when n equals m, the horizontal asymptote equals y = a/b.
  • There may be no horizontal asymptote when n is more than m.

The stages of the polynomials inside the feature determine whether or not a horizontal asymptote exists. In addition, it also determines where it is located. Let’s examine how we can apply those criteria to find our horizontal asymptotes.

Horizontal asymptotes are just good recommendations, but vertical asymptotes are hallowed territory. You should never come into contact with a vertical asymptote. You can touch, and even move, horizontal asymptotes.

Read Also: Linear Interpolation Formula

Vertical asymptotes indicate extremely specific behavior (at the graph), usually near the origin. Horizontal asymptotes indicate popular behavior, which is usually far away from the graph’s edges. The horizontal asymptotes differ from vertical asymptotes in several significant ways.

                                                                          Horizontal Asymptotes

Horizontal Asymptotes of Exponential Functions

Variables can be exponents. Moreover, variable exponents obey all of the exponent characteristics stated in Properties of Exponents. An exponential function is one that has a changeable exponent. Exponential functions include f (x) = 2x and g(x) = 53x. We can draw graphs of exponential functions.

However, we can also graph exponential functions with other bases, like f (x) = 3x and f (x) = 4x. These graphs are differing from the graph of f (x) = 2x by a horizontal stretch or shrink. When we multiply the input of f (x) = 2x by 2, we get f (x) = 22x = (22)x = 4x. Therefore, the graph of f (x) = 4x is shrunk horizontally by a factor of 2 from f (x) = 2x:

Oblique Horizontal Asymptote or Slant Asymptote

Some curves feature oblique asymptotes, which are neither horizontal nor vertical.

If the vertical distances between the curve y = f(x) and the line y = MX + b approach zero, then the line y = MX + b is termed the oblique or slant asymptote.

Oblique asymptotes arise for rational functions when the degree of the numerator is one more than the degree of the denominator. We also may use the long division to find the equation of the oblique asymptote in this situation.

Horizontal Asymptote Example

Find the horizontal asymptote of (x2−4x+5)x−6(x2−4x+5)x−6.

Solution:

Here, the vertical asymptote occurs at x=6.

Horizontal Asymptotes

There is no horizontal asymptote since the degree of the numerator is greater than the denominator.

What are the 3 types of asymptotes?

There are in total three types of asymptotes, they are horizontal, vertical and oblique.

How do you find the horizontal asymptote of a function?

If you have an equation with numerator and denominator then:

Deg N(x) = the degree of a numerator.

Deg D(x) = the degree of a denominator.

Now divide N(x) by D(x).

Let y= N(x)/D(x).

If the value of y is constant, then it is an equation of a horizontal asymptote.

Can a function have more than one horizontal asymptote?

A function can have a maximum of two different horizontal asymptotes.

Can a rational function have two horizontal asymptotes?

A rational function can have only on horizontal asymptote at most.

Can a function cross its horizontal asymptote?

Yes, a graph of a function can definitely cross its horizontal asymptote.